Final answer:
By setting up the mean and standard deviation equations for the data set {12,14,19,x}, and solving them, we find that x = 15 is the value that satisfies both the required mean of 15 and the standard deviation of √26.
Step-by-step explanation:
To find the value of x when the mean and standard deviation of the data set {12,14,19,x} are 15 and √26 respectively, we can set up two equations based on the definitions of mean and standard deviation:
- The mean equation is given by (12 + 14 + 19 + x) / 4 = 15.
- The standard deviation equation involves the mean, each data point, and finding the square root of the variance.
Since we already have the mean, we plug in our values: (12 + 14 + 19 + x) = 15 * 4
which simplifies to 45 + x = 60. Solving for x gives us x = 15.
However, we also need to make sure that this value of x gives us the correct standard deviation. For that, the variance (the square of the standard deviation) will be the mean of the squared differences between each data value and the mean. Since our required standard deviation is √26, our variance needs to be 26.
We calculate the squared differences for known values: (12-15)² + (14-15)² + (19-15)² + (x-15)², and find their average is √26.
Simplifying and solving this will confirm that x = 15 is the correct value that also satisfies the standard deviation requirement.